4. HALO CENTERS

Already in the early 1990s, high resolution simulations of individual
galaxy halos in CDM were finding
(r) ~
r- with
~ 1. This behavior
implies that the rotation velocity at
the centers of galaxies should increase as r1/2, but
the data,
especially that on dark-matter-dominated dwarf galaxies, instead
showed a linear increase with radius, corresponding to roughly
constant density in the centers of galaxies. This disagreement of
theory with data led to concern that CDM might be in serious
trouble [65,
66].

Subsequently, NFW
[58]
found that halos in all variants of CDM are well fit by the
NFW(r) given above, while Moore's group
proposed an alternative
(r) r-3/2(r + rM)-3/2
based on a small number of very-high-resolution simulations of
individual halos
[67,
68,
69].
Klypin and collaborators (including me) initially claimed that typical
CDM halos have shallow inner profiles with
0.2
[72],
but we subsequently realized that the convergence tests that we had
performed on these simulations were inadequate. We now have simulated
a small number of galaxy-size halos with very high resolution
[59],
and find that they range between
NFW and
M. Actually, these two analytic density profiles
are almost indistinguishable unless galaxies are probed at scales
smaller than about 1 kpc, which is difficult but sometimes possible.

Meanwhile, the observational situation is improving. The rotation
curves of dark matter dominated low surface brightness (LSB) galaxies
were measured with radio telescopes during the 1990s, and the rotation
velocity was typically found to rise linearly at their centers
[70,
71,
72].
But a group led by van den Bosch
[73]
showed that in many cases the large beam size of the
radio telescopes did not adequately resolve the inner parts of the
rotation curves, and they concluded that after correcting for beam
smearing the data are on the whole consistent with expectations from
CDM. Similar conclusions were reached for dwarf galaxies
[74].
Swaters and collaborators showed that optical
(H) rotation curves of
some of the LSB galaxies rose
significantly faster than the radio (HI) data on these same galaxies
[75]
(see Fig. 2), and these rotation curves (except for
F568-3) appear to be more consistent with NFW
[76]. At a
conference in March 2000 at the Institute for Theoretical Physics in
Santa Barbara, Swaters also showed a
H rotation curve for the
nearby dwarf galaxy DDO154, which had long been considered to be a
problem for CDM
[65,
66];
but the new, higher-resolution data appeared consistent with an inner
density profile 1.
(5)

Very recently, a large set of high-resolution optical rotation curves
has been analyzed for LSB galaxies, including many new observations
[77].
The first conclusion that I reach in looking at the
density profiles presented is that the NFW profile often appears to be
a good fit down to about 1 kpc. However, some of these galaxies
appear to have shallower density profiles at smaller radii. Of the 48
cases presented (representing 47 galaxies, since two different data
sets are shown for F568-3), in a quarter of the cases the data do not
probe inside 1 kpc, and in many of the remaining cases the resolution
is not really adequate for definite conclusions, or the interpretation
is complicated by the fact that the galaxies are nearly edge-on. Of
the dozen cases where the inner profile is adequately probed, about
half appear to be roughly consistent with the cuspy NFW profile (with
fit 0.5), while half
are shallower. This is not
necessarily inconsistent with CDM, since observational biases such as
seeing and slight misalignment of the slit lead to shallower profiles
[78].
Perhaps it is significant that the cases where
the innermost data points have the smallest errors are cuspier.

I think that this data set may be consistent with an inner density
profile ~ 1 but
probably not steeper, so it is definitely
inconsistent with the claims of the Moore group that
1.5. But very
recent work by Navarro and collaborators
[79]
has shown that Moore's simulations did
not have adequate resolution to support their claimed steep central
cusp; the highest-resolution simulations appear to be consistent with
NFW, or even shallower with
0.75. Further
simulations and observations, including measurement of CO rotation
curves
[80],
may help to clarify the nature of the dark matter.

It is something of a scandal that, after all these years of simulating
dark matter halos, we still do not have a quantative - or even a
qualitative - theory explaining their radial density profiles. In
her dissertation research
[63],
Risa Wechsler found that the
central density profile and the value of rs are typically
established during the early, rapidly merging phase of halo evolution,
and that, during the usually slower mass accretion afterward,
rs
changes little. The mass added on the halo periphery increases
Rvir, and thus the concentration
cvirRvir/rs.
Now we want to understand this analytically. Earlier attempts to
model the result of sequences of mergers (e.g.,
[81,
82])
led to density profiles that depend strongly on the
power spectrum of initial fluctuations, in conflict with simulations
(e.g. [83]).
Perhaps it will be possible to improve on the
simple analytic model of mass loss due to tidal stripping during
satellite inspiral that we presented in
[101].
Avishai Dekel and his students have recently shown that including the tidal
puffing up of the inspiralling satellite before tidal stripping can
perhaps account for the origin of the cusp seen in dissipationless
simulations, independent of the power spectrum. They argue that the
profile must be steeper than
= 1 as long as enough
satellites make it into the halo inner regions, simply because for flatter
profiles the tidal force causes dilation rather than stripping. The
proper modeling of the puffing and stripping in the merger process of
CDM halos may also provide a theoretical framework for understanding
the observed flat cores as a result of gas processes; work on this by
Ari Maller and Dekel is in progress. Reionization and feedback into
the baryonic component of small satellites would make their cores puff
up before merging. This could cause them to be torn apart before they
penetrate into the halo centers, and thus allow
< 1 cores.

Another possible explanation for flatter central density profiles
involving the baryonic component in galaxies has recently been proposed
[84],
in which the baryons form a bar that
transfers angular momentum into the inner parts of the halo. It is
not clear, however, that this effect could be very important in dark
matter dominated dwarf and LSB galaxies that have small or nonexistent
bulge components.

It would be interesting to see whether CDM can give a consistent
account of the distribution of matter near the centers of big
galaxies, but this is not easy to test. One might think that big
bright galaxies like the Milky Way could help to test the predicted
CDM profile, but the centers of such galaxies are dominated by
ordinary matter (stars) rather than dark matter.
(6)

5 Swaters (private communication)
and Hoffman have subsequently confirmed this with better data, which
they are preparing for publication.
Back.

6 Navarro and
Steinmetz had claimed that the Milky Way is inconsistent with the NFW
profile [85],
but they have now shown that
CDM simulations
with a proper fluctuation spectrum are actually consistent with the data
[86].
Back.